Micron-level accuracy over a meter?

[RJSakowski]

I would still try to use an optoisolator similar to what I had used. The flag can be a piece of thin opaque plastic mounted so the plane of the flag is perpendicular to a line through the pivot point. A wider beam is OK for detecting a vehicle passing but is not very precise as far as the trigger point is concerned. You really want a nice crisp trigger for optimum repeatability. Considering that you will be adjusting the distance between the knife edges, the sensor position will need to be adjustable in order to accommodate the difference in distance between the flag and the knife edge. You could use two flags. A thin piece of plastic will contribute only a insignificant amount to the center of oscillation of the pendulum. and air resistance.

Were this my project, I would start out by not worrying too much about a high degree of accuracy. I would concentrate on making the right mechanism and on getting the timing circuitry right. Once I established that I could get a reasonable value for g, then I would look into what I could do to improve the precision of the number. Some things to consider; environmental controls to deal with temperature fluctuations, isolation of the mechanism to prevent disturbance from passing vehicles (and what about earth tremors?), possibly operating in a vacuum.

As to measuring the distance between the knife edges, I think I would apporoach it like this. After making the adjustments to the knife edge position to equalize the periods, I would make a fixture to hold the pendulum and support a rod to fit between the knife edges. I would make a rod to fit short of the touching both knife edges and use feeler gages to make up the difference. The thickness of the feeler gage pack could then be measured with a micrometer and added to the rod length to determine the distance between the knife edges. I would then have the rod calibrated buy a metrology lab or any other convenient means, depending on my desired accuracy. This should give accuracy to a few ppm. Anither way would be to make the test rod about an inch short of the distance and make pins to make up the difference. It wouldn't take much effort to make a set of pins, differing by .0001" and choosing the best fit. The pins could be calibrated at the same time as the test rod.

 

[zondar]

Thank you for your feedback.

I've thought a little about making a historical replica of Kater's pendulum, designed from photographs of an original and the historical records, with period materials, etc., if only as a cool "art" object rather than a practical one. But that raises the difficulty level considerably. In one extreme example, the blades were made of wootz, which was the best steel available at the time, but now is effectively a lost technology.

Instead, I'm planning on making a Repsold-Bessel pendulum. It was an improvement on Kater's pendulum in that it's designed to be perfectly symmetrical, except with a heavy weight at one end and a dummy weight at the other (as light as practical but in the exact same shape as the heavy weight for the sake of symmetry, i.e. to match air resistance).

This pendulum does not need to have the knife edges be adjustable, at least if you aren't aiming at a given period (e.g. a precise second's pendulum), and it also doesn't need a sliding weight to match the periods when the pendulum is reversed. Kater needed his pendulum to be almost exactly a second due to the way he measured the period, but with with modern timing equipment an arbitrary period will do.

Since the knives would be rigidly mounted in this variant, I'd mostly expect a metrology lab to be able to measure L directly, e.g. on a traveling microscope. But I don't really know what the reaction or difficulties might be.

I'm an EE and am familiar with electro-optical circuits. I have an oscilloscope and could borrow a very accurate counter. I'd expect the trigger circuits on them to be decent enough for a first try, and I can handle the rest.

For the flags, I was thinking of two thin metal rods (just for simplicity in machining them in on a lathe), mounted to both ends past the knife holders. This is similar to what Kater and Repsold used.

Having the knives and anvil (that the knives would ride on) be very straight and perpendicular to the swing would be very important or the pendulum will wobble, destroying its accuracy. Similarly, everything else needs to be symmetrical and in-line.

But I'm such a beginner at machining that I'm probably not capable of high precision! My tools are severely lacking too. Anyway, this project is about the fun of learning. I intend to start with an inexpensive prototype (i.e., not made of Invar) that I won't feel bad about messing up, and yet is still capable of testing the whole system.

I live in suburbia and there is constant geological noise from cars and trucks driving around, etc. I'd have to rely on averaging to wash out some of that.

As for a vacuum, ha ha, I might be crazy enough to make a project out of this, but I'm not so crazy as to go that far! (But yes, Kater also used his in a partial vacuum at some point.)

Thanks again to everyone for the interesting and useful comments.

 

[rwm]

This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.

 

[RJSakowski]

This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.

Kater's pendulum - Wikipedia

en.wikipedia.org

 

[rwm]

Thanks RJ, I read that but it did not really answer the question or I missed it? I did see this:
"In Kater's time, the period T of pendulums could be measured very precisely by timing them with precision clocks set by the passage of stars overhead. "
What were these precision clocks? Were these like the automatic movements I have on my wrist today? I assume he transported these around with the pendulum? I am trying to understand what they used as a time base that was not also affected by gravity.

 

[zondar]

This is fascinating. May I ask a question about this experiment? As you indicate, you can easily measure the period of the pendulum with modern electronics. How did Kater do it? Did he have a standardized clock that he carted around with him? I assume gravity would not affect a precision watch like an automatic movement? That relies on mass but not gravity.

He compared a calibrated pendulum clock (calibrated against the movement of stars) against the test pendulum positioned so that they overlap visually. He observed the two overlapping pendulums via an optical telescope from across the room to avoid parallax errors.

Both were almost exactly in sync, but not quite. Then he watched for coincidences between the two pendulums (the clock's and the test pendulum). For illustration, if every hundred swings the two pendulums were in alignment, then the two periods differ by 1/100th of the calibrated clock's period (I haven't done the math, but something like that). From that, he can calculate the period of the test pendulum to very high accuracy. He called this the "method of coincidences."

This is similar to how a vernier caliper works.

 

[rwm]

Thank you! That is part of what I don't understand. So the Kater pendulum is affected by gravity but the simple pendulum is not?

 

[zondar]

Thank you! That is part of what I don't understand. So the Kater pendulum is affected by gravity but the simple pendulum is not?

Both pendulums are affected by gravity. The reference pendulum clock is calibrated against the stars so its pendulum has an accurately-known period of oscillation.

You might ask, why not just use that calibrated pendulum to measure the force of gravity? Well, you can't, because an ordinary physical pendulum is a so-called "compound pendulum." For example, the wood or metal rod holding it has mass too, not just the heavy bob, and that messes up the calculation. Even using a very thin metal wire is too much, plus that stretches by different amounts during swings, etc.

Another reason why you can't use the clock's pendulum to measure gravity is that the clock adds energy to the pendulum's swing with each beat (or else it would stop). That turns out to mess up everything even more.

So Kater devised a special pendulum that, after adjustment by reversing it until both orientations have the same period, does act like an ideal pendulum, making it amenable to the gravity calculation. But now you have to figure out what its period is (and its length). So Kater compared its swings against the reference pendulum by the method of coincidences.

 

[rwm]

I think something just clicked that was not directly explained. I assumed that the clock was built at the same location as the Kater's pendulum and calibrated there. That would mean they were calibrated in the same gravity and should always read the same. Now if the clock was calibrated against the stars at another specific location and then Kater's pendulum was brought to that location they could swing with different periods. Is that the missing piece?
I do understand the method of coincidences and how it is like a vernier. That makes perfect sense.

 

[zondar]

I think something just clicked that was not directly explained. I assumed that the clock was built at the same location as the Kater's pendulum and calibrated there. That would mean they were calibrated in the same gravity and should always read the same. Now if the clock was calibrated against the stars at another specific location and then Kater's pendulum was brought to that location they could swing with different periods. Is that the missing piece?
I do understand the method of coincidences and how it is like a vernier. That makes perfect sense.

That is true: The reference clock must be calibrated (though not necessarily built) at the same location, or else the reference clock will become inaccurate once moved, since the force of gravity depends on where you are located, and that will change the period of oscillation of the reference clock.

Then the Kater's pendulum just needs to be in the same location as the stationary reference clock. He did actually use at least two different reference clocks, and you see two in that engraving, one behind the Kater's pendulum for the coincidence measurement, and one to the side.